The Stacks project

Lemma 20.25.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be an open covering. Let $\mathcal{F}^\bullet $ be a bounded below complex of $\mathcal{O}_ X$-modules. If $H^ i(U_{i_0 \ldots i_ p}, \mathcal{F}^ q) = 0$ for all $i > 0$ and all $p, i_0, \ldots , i_ p, q$, then the map $ \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \to R\Gamma (X, \mathcal{F}^\bullet ) $ of Lemma 20.25.1 is an isomorphism.

Proof. Immediate from the spectral sequence of Lemma 20.25.1. $\square$

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